The global symmetries of spin systems defined on abelian groups. I

Abstract

We consider the classification problem of the global symmetry groups of spin systems defined on abelian groups. Its implications on the generating functional, the transfer matrix, the Hamiltonian formalism, and factorization properties of spin systems are discussed. The duality properties of spin systems defined on semidirect products of abelian groups are revisited. In the first of this series of three papers we list the groups for systems defined on Zp (p prime), Z2⊗Z2, and Z2⊗Z2⊗Z2 manifolds. They are direct or wreath products of M-metacyclic groups and symmetric groups.